A schematic illustration of the IceCube experiment is shown in Fig. REF . Neutrinos are detected through the
Cherenkov light emitted by secondary particles produced
in neutrino-nucleon interactions in or around the detector. Although primarily designed for the detection of high-energy
neutrinos from astrophysical sources, denoted cosmic neutrino in Fig. REF , IceCube can also be
used for investigating the atmospheric neutrino spectrum.
The atmospheric neutrinos are produced in cosmic-ray interactions with nuclei
in Earth's atmosphere [1]}. While at low neutrino
energies (\(E_{\nu }\lesssim 10^5\)  GeV), these neutrinos arise from the
decay of light mesons (pions and kaons), and the associated flux is
denoted as the conventional atmospheric neutrino flux
[2]}, for larger energies
it is expected that the prompt atmospheric neutrino flux associated with the
decay of hadrons containing heavy flavours become important
[3]}.
One has that the flux of conventional atmospheric neutrinos is a function
of the zenith angle, since horizontally travelling mesons have
a much higher probability to decay before losing energy in
collisions, which implies a harder conventional neutrino spectrum of
horizontal events compared to vertical events. In contrast, heavy mesons decay before interacting and follow the initial
spectrum of cosmic rays more closely, being almost independent of the zenith angle in the neutrino energy range probed by the IceCube.
As discussed in the Introduction,
the calculation of the prompt atmospheric neutrino flux at the
detector level depends on the description of the production and decay of the
heavy hadrons as well as the propagation of the associated particles
through the atmosphere (see Fig. REF ). Following our
previous studies [4]}, [5]}, we will
estimate the expected prompt neutrino flux in the detector
\(\phi _{\nu }\)  using the \(Z\) -moment method [3]}, which implies
that \(\phi _{\nu }\)  can be estimated using the geometric interpolation formula
\(\phi _{\nu } = \sum _H \frac{\phi _{\nu }^{H,low} \cdot \phi _{\nu }^{H,high}}{\phi _{\nu }^{H,low} + \phi _{\nu }^{H,high}} \,\,.\) 
